ADAPTATION OF THE REYNOLDS STRESS ...1Laboratoire deMécanique des Fluideset d’Acoustique,Ecole Centrale Lyon, France 2NUMTECH, France 2 Outline 1. Introduction and motivations 2

ADAPTATION OF THE REYNOLDS STRESS TURBULENCE MODEL FOR ATMOSPHERIC SIMULATIONS

Radi Sadek1, Lionel Soulhac1, Fabien Brocheton2 and Emmanuel Buisson2

15th Harmonisation within Atmospheric Dispersion Modelling for Regulatory Purposes Conference

1Laboratoire de Mécanique des Fluides et d’Acoustique, Ecole Centrale de Lyon, France2NUMTECH, France

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Outline

1. Introduction and motivations

2. Reynolds Stress model

3. Atmospheric RSM constants

4. Validation of the model

5. Conclusions and perspectives

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1 – Introduction and motivations

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1 – Introduction and motivationsResearch issues in atmospheric CFD modeling

• Atmospheric CFD simulations (particularly RANS k‐) are often used• For local scale wind engineering

• For dispersion in complex urban area with obstacles

• For risk and safety assessment in industrial areas

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1 – Introduction and motivationsResearch issues in atmospheric CFD modeling

• Different issues have to be properly solved• Boundary conditions over the surface layer?

• Anisotropy of the turbulence?

• “Standard” or “atmospheric/Duynkerke” constants?

• Value of the turbulent Schmidt number?

Large uncertainty and user‐dependent variability when comparing with field measurements

• Objectives of this work• Introduce anisotropy of turbulence using Reynolds Stress Model

• Develop a 1D model for the entire Atmospheric Boundary Layer

Provide parameterizations and boundary conditions for 3D CFD calculations

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2 – Reynolds Stress model

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2 – Reynolds Stress modelRSM equations

• Reynolds Stress Model equations• Reynolds Stress equation

• Turbulent dissipation rate equation

• Turbulent viscosity

i j i j i jmk ij ij ij

k k k k

j iij i k j k

k k

uu uu uuK 2u P

t x x x 3

u uwith P uu uu

x x

2m

i 1 ii 2i k k

Ku C P C

t x x x 2k 2k

2

mk

K C

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2 – Reynolds Stress modelRSM constants

• The preceding equations depends on 5 constants

• Pressure‐strain term ij• We choose the model of Gibson and Launder (1978)

• This model introduces 5 other constants :

• The “standard” values of these constants are:

k 1 2C , , , C , C

1 2 1 2 LC , C , C , C , C

C k e C1 C2 C1 C2 C’1 C’2 CL0.09 1.0 1.3 1.44 1.92 1.8 0.6 0.5 0.4 0.39

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2 – Reynolds Stress modelRSM constants

• “Standard” constants are not adapted for the atmosphere

• Consider the Surface Boundary Layer• For example, the C constant controls the level of turbulent

kinetic energy k:

• Wind tunnel measurements give B = 3.33 (i.e. C = 0.09)

• Atmospheric measurements (Panofsky and Dutton, 1984) give B = 5.48 (i.e. C = 0.033)

• It is necessary to define “atmospheric” constants:• Duynkerke (1988) proposed a set of constants for the k‐

model

• In this work, we propose a new set of constants for the Reynolds Stress Model

2*

k 1B

u C

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3 – Atmospheric RSM constants

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3 – Atmospheric RSM constantsDetermination approach

• In the Atmospheric Surface Layer• We assume a 1D Atmospheric Surface Layer, horizontally

uniform

• We identify the 1D RSM equations with the following analytical profiles

*

0

3*

2 2 2x *

2 2 2y *

2 2 2z *

2*

u zu z ln

z

uz

z

u u

v u

w u

u w u

x

y

z

2.461.9

1.17

with (Panofsky and Dutton, 1984)

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3 – Atmospheric RSM constantsDetermination approach

• It provides a system of equations for the constants:

• Which provides, after resolution, a set of “atmospheric” constants:

1 322 2x 2 2

1 1 L z 2 2 L 1*

2 1 3y2 22 2

1 1 L z 2 2 L 1*

1 322 2z 2 2

1 1 L z 2 2 L

2 z u 2u equation C C C B C 2 C B C 2 C 1 0

B 3 u z 3

2 z u 2v equation C C C B C 1 C B C C 1 0

B 3 u z 3

2w equation C 2C C B C 1 2C B C

B 3

1*

322

1 1 2 2 L z*

z u 2C 1 0

u z 3

1 3 3 z uu w equation C C C 1 C B C 1 0

B 2 2 u z

C k e C1 C2 C1 C2 C’1 C’2 CL0.033 1.0 2.38 1.46 1.83 1.8 0.6 0.94 0.03 0.19

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4 – Validation of the model

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4 – Validation of the modelMethodology of the 1D numerical model

• We have developped a numerical 1D model for the ABL• Flow is horizontally homogenous:

• Pressure gradient and Coriolis force in geostrophic balance

• 1D equation model of mean wind speed, including Coriolis effect:

• Reynolds Stress Model as turbulence closure model

• Equations are solved numerically until a steady state

• We apply the model on the overall ABL and compare it with empirical results

g gu u w v v w

f v V and f u Ut z t z

w 0 and 0

x y

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4 – Validation of the modelResults in neutral conditions

• Velocity profile:• For example for Ug =5m.s−1, z0 =0.01m, = 45°

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• Turbulence profiles (normalized by the ground value):• For example for Ug =5m.s−1, z0 =0.01m, = 45°

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• Ekman’s theory predicts the twisting of the flow in the Ekman layer:

• Simulation of the Ekman Layer with the 1D model:

g

Mg

u U 1 exp az cos az f with a

2Kv sgn f U exp az sin az

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• The Rossby similarity theory gives:

• Sensibility of the 1D model to control parameters Ug, f and z0

A and B are empirical constants

2

g 2*

* 0

U u1ln B A

u fz

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• Atmospheric Boundary Layer height:• The Rossby‐Montgomerry equation

*cu

h with c 0.26f

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4 – Validation of the modelResults in stratified conditions

• Velocity profile:• For example for Ug =5m.s−1, z0 =0.01m, = 45°

Unstable conditions Stable conditions

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Unstable conditions

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Stable conditions

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• Relation between Ug and u*:

A and B are dependent on stability, measured empirically

Unstable conditions Stable conditions

2

g 2*

* 0

U u1ln B A

u fz

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• The Rossby similarity theory gives:

A is dependent on stability, measured empirically

Surface wind

Geostrophic windα

*

g

uAsin sign f

U

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5 – Conclusions and perspectives

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5 ‐ Conclusions and perspectives

• Conclusions• Development of a new set of atmospheric constants for

the Reynolds Stress turbulence model

• 1D simulations of the Atmospheric Boundary Layer with this Reynolds Stress Model, including Coriolis effects

• Validation against empirical results

• Perspectives• Validation in more complex configurations

• Evaluation of the effect of anisotropy on dispersion modeling

• Unified approach between “atmospheric” and “standard” RSM constants (see Poster H15‐166)

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Thank you for your attention

Questions ?

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ADAPTATION OF THE REYNOLDS STRESS ...1Laboratoire deMécanique des Fluideset d’Acoustique,Ecole Centrale Lyon, France 2NUMTECH, France 2 Outline 1. Introduction and motivations 2